A mass conserving mixed stress formulation for the Stokes equations. (17th May 2019) - Gopalakrishnan, Jay; Lederer, Philip L; Schöberl, Joachim
Abstract: We propose stress formulation of the Stokes equations. The velocity u is approximated with H(div) -conforming finite elements providing exact mass conservation. While many standard methods use H^1 -conforming spaces for the discrete velocity H(div) -conformity fits the considered variational formulation in this work. A new stress-like variable σ equalling the gradient of the velocity is set within a new function space H(curl div) . New matrix-valued finite elements having continuous 'normal-tangential' components are constructed to approximate functions in H(curl div) . An error analysis concludes with optimal rates of convergence for errors in u (measured in a discrete H^1 -norm), errors in σ (measured in L^2 ) and the pressure p (also measured in L^2 ). The exact mass conservation property is directly related to another structure-preservation property called pressure robustness, as shown by pressure-independent velocity error estimates. The computational cost measured in terms of interface degrees of freedom is comparable to old and new Stokes discretizations.